A221371 - OEIS

%I #13 Sep 16 2023 06:49:42

%S 1,1,4,23,209,2744,49539,1180281,35921892,1360513711,62770245601,

%T 3466178083312,225719029475675,17117740162448105,1495526385479298140,

%U 149120758170390404103,16831018302445533666705,2134813624482300873515304,302332062412598445891728563

%N O.g.f.: Sum_{n>=0} n!^2 * x^n * Product_{k=1..n} (1 + x) / (1 + k^2*x + k^2*x^2).

%C Compare to the o.g.f. of A110501, the unsigned Genocchi numbers (of first kind):

%C Sum_{n>=0} n!^2 * x^(n+1) / Product_{k=1..n} (1 + k^2*x).

%H Vaclav Kotesovec, <a href="/A221371/b221371.txt">Table of n, a(n) for n = 0..240</a>

%F O.g.f.: A(x) = 1/(1-x*(1+x)/(1-2*x/(1-4*x*(1+x)/(1-6*x*(1+x)/(1-9*x*(1+x)/(1-12*x*(1+x)/(... -[(n+1)/2]*[(n+2)/2]*x*(1+x)/(1- ...)))))))) (continued fraction).

%F a(n) = Sum_{k=0..[n/2]} binomial(n-k,k) * A110501(n+1), where A110501(n) = 2*(-1)^n*(1-4^n)*B_{2*n} (B = Bernoulli numbers).

%F a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (exp(2*n) * Pi^(2*n+3/2)). - _Vaclav Kotesovec_, Nov 02 2014

%e O.g.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 209*x^4 + 2744*x^5 + 49539*x^6 + ...

%e where

%e A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!^2*x^2*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)) + 3!^2*x^3*(1+x)*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)*(1+9*x+9*x^2)) + 4!^2*x^4*(1+x)*(1+x)*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)*(1+9*x+9*x^2)*(1+16*x+16*x^2)) + ...

%t a[n1_Integer?NonNegative, n2_Integer?NonNegative] := CoefficientList[Sum[(m!)^2*x^m*Product[(1 + x)/(1 + k^2*x + k^2*x^2), {k, 1, m}], {m, 0, n2 + 1}] + O[x]^(n2 + 2), x][[n1 + 1 ;; n2 + 1]]; a[0, 18] (* _Robert P. P. McKone_, Sep 16 2023 *)

%o (PARI) {a(n)=polcoeff( sum(m=0, n, m!^2*x^m*prod(k=1, m, (1+x)/(1+k^2*x+k^2*x^2 +x*O(x^n))) ), n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=sum(k=0,n\2,binomial(n-k,k)*2*(-1)^(n-k+1)*(1-4^(n-k+1))*bernfrac(2*(n-k+1)))}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A221370, A110501.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 13 2013